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Naturalness, Weak Scale Supersymmetry and the Prospect

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Naturalness, Weak Scale Supersymmetry and the Prospect Naturalness, Weak Scale Supersymmetry and the Prospect for the Observation of Supersymmetry at the Tevatron and at the LHC Kwok Lung Chan, Utpal Chattopadhyay and Pran Nath Department of Physics, Northeastern University Boston, MA 02115-5005 Abstract Naturalness bounds on weak scale supersymmetry in the context of radiative break- ing of the electro-weak symmetry are analyzed. In the case of minimal supergravity it is found that for low tanβ and for low values of fine tuning Φ, where Φ is defined 2 2 essentially by the ratio µ /MZ where µ is the Higgs mixing parameter and MZ is the Z boson mass, the allowed values of the universal scalar parameter m0, and the universal gaugino mass m1/2 lie on the surface of an ellipsoid with radii fixed by Φ leading to tightly constrained upper bounds √Φ. Thus for tanβ 2( 5) it is found that the ∼ ≤ ≤ upper limits for the entire set of sparticle masses lie in the range < 700 GeV (< 1.5TeV) for any reasonable range of fine tuning (Φ 20). However, it is found that there exist ≤ regions of the parameter space where the fine tuning does not tightly constrain m0 and m1/2. Effects of non-universalities in the Higgs sector and in the third generation sector on naturalness bounds are also analyzed and it is found that non-universalities can significantly affect the upper bounds. It is also found that achieving the maximum Higgs mass allowed in supergravity unified models requires a high degree of fine tuning. arXiv:hep-ph/9710473v3 24 Jun 1998 Thus a heavy sparticle spectrum is indicated if the Higgs mass exceeds 120 GeV. The prospect for the discovery of supersymmetry at the Tevatron and at the LHC in view of these results is discussed. 1 Introduction One of the important elements in supersymmetric model building is the issue of the mass scale of the supersymmetric particles. There is the general expectation that this scale should be of the order of the scale of the electro-weak physics, i.e., in the range of a TeV. This idea is given a more concrete meaning in the context of supergravity unification[1] where one has spontaneous breaking of the electro-weak symmetry by radiative corrections[2]. Radiative breaking of the electro-weak symmetry relates the scale of supersymmetry soft breaking terms directly to the Z boson mass. This rela- tionship then tells us that the soft SUSY breaking scale should not be much larger 1 than the scale of the Z boson mass otherwise a significant fine tuning will be needed to recover the Z boson mass. The above general connection would be thwarted if there were large internal cancellations occurring naturally within the radiative breaking con- dition which would allow m0 and m1/2 disproportionately large for a fixed fine tuning. We shall show that precisely such a situation does arise in certain domains of the supergravity parameter space. The simplest fine tuning criterion is to impose the constraint that m0,mg˜ < 1 TeV where m0 is the universal soft SUSY breaking scalar mass in minimal supergravity and mg˜ is the gluino mass. The above criterion is easy to implement and has been used widely in the literature (for a review see Ref.[3]). A more involved fine tuning criterion is given in Ref.[4]. However, it appears that the criterion of Ref.[4] is actually a measure of the sensitivity rather than of fine tuning[5, 6]. Another naturalness criterion is proposed in Ref.[6] and involves a distribution function. Although the distribution function is arbitrary the authors show that different choices of the function lead numerically to similar fine tuning limits. In the analysis of this paper we use the fine tuning criterion introduced in Ref.[7] in terms of the Higgs mixing parameter µ which has several attractive features. It is physically well motivated, free of ambiguities and easy to implement. Next we use the criterion to analyze the upper limits of sparticle masses for low values of tanβ, i.e., tanβ 5. In this case one finds that m0 and m1 2 allowed by radiative breaking lie ≤ / on the surface of an ellipsoid, and hence the upper limits of the sparticle masses are directly controlled by the radii of the ellipsoid which in turn are determined by the choice of fine tuning. For instance, one finds that if one is in the low tanβ end of b τ − unification[8] with the top mass in the experimental range, i.e. tanβ 2, then for ≈ any reasonable range of fine tuning the sparticle mass upper limits for the entire set of SUSY particles lie within the mass range below 1 TeV. Further, one finds that the light Higgs mass lies below 90 GeV under the same constraints. Thus in this case discovery of supersymmetry at the LHC is guaranteed according to any reasonable fine tuning criterion. Next the paper explores larger values of tanβ, i.e., tanβ 10 and here one ≥ finds that m0 and m1/2 for moderate values of fine tuning do not lie on the surface of an ellipsoid; rather one finds that they lie on the surface of a hyperboloid. In this case m0 and m1/2 are not bounded by the µ constraint equation and large values of m0 and m1/2 can result with a fixed fine tuning. Effect of non-universalities on naturalness is also analyzed. Again one finds phe- nomena similar to the ones discussed above, although the domains in which these phe- nomena occur are shifted relative to those in the universal case. One of the important results that emerges is that the upper limits of sparticle masses can be dramatically affected by non-universalities. These results have important implications for the dis- covery of supersymmetry at the Tevatron and the LHC. Our analysis is carried out in the framework of supergravity models with gravity mediated breaking of supersymmetry[9, 1, 3]. This class of models possesses many attractive features. One of the more attractive features of these models is that with R parity invariance the lightest neutralino is also the lightest supersymmetric particle over most of the parameter space of the theory and hence a candidate for cold dark matter. Precision renormalization group analyses show[10] that these models can accommodate 2 just the right amount of dark matter consistent with the current astrophysical data[11, 12]. However, in this work we shall not impose the constraint of dark matter. The outline of the paper is as follows: In Sec.2 we give a brief discussion of the fine tuning measure used in the analysis. In Sec.3 we use this criterion to discuss the upper limits on the sparticle masses in minimal supergravity for low tanβ, i.e., tanβ 5 and show that the allowed solutions to radiative breaking lie on the surface ≤ of an ellipsoid. In Sec.4 we discuss naturalness in beyond the low tanβ region. Here we show that radiative breaking of the electro-weak symmetry leads to the soft SUSY breaking parameters lying on the surface of a hyperboloid. In Sec.5 we discuss the effects of non-universalities on the upper limits. In Sec.6 we show that a high degree of fine tuning is needed to have the light Higgs mass approach its maximum upper limit. The limits on Φ from the current data are discussed in sec.7. Implications of these results for the discovery of supersymmetric particles at colliders is also discussed in Secs. 3-6. Conclusions are given in Sec.8. 2 Measure of Naturalness We give below an improved version of the analysis of the fine tuning criterion given in Ref.[7]. The radiative electro-weak symmetry breaking condition is given by 1 2 2 2 M = λ µ (1) 2 Z − where λ2 is defined by 2 2 2 2 m¯ m¯ tan β λ = H1 − H2 (2) tan2 β 1 − 2 2 Herem ¯ Hi = mHi + Σi(i=1,2) where Σi arise from the one loop corrections to the effective potential[13]. The issue of fine tuning now revolves around the fact that a cancellation is needed between the λ2 term and the µ2 term to arrange the correct 2 experimental value of MZ . Thus a large value of λ would require a large cancellation from the µ2 term resulting in a large fine tuning. This idea can be quantified by defining the fine tuning parameter Φ so that 2 2 −1 λ µ Φ = 4 − (3) λ2 + µ2 (The factor of 4 on the right hand side in Eq.(3) is just a convenient normalization.) The expression for Φ can be simplified by inserting in the radiative breaking condition Eq.(1). We then get 1 µ2 Φ= + 2 (4) 4 MZ The result above is valid with the inclusion of both the tree and the loop corrections to the effective potential.(Φ is related to the fine tuning parameter δ defined in Ref.[7] −1 µ2 by Φ = δ ). For large µ one has Φ M 2 , a result which has a very direct intuitive ∼ Z meaning. A large µ implies a large cancellation between the λ2 term and the µ2 term 3 in Eq.1 to recover the Z boson mass and thus leads to a large fine tuning. Typically a large µ implies large values for the soft supersymmetry breaking parameters m0 and m1/2 and thus large values for the sparticle masses. However, large cancellation can be enforced by the internal dynamics of radiative breaking itself. In this case a small µ and hence a small fine tuning allows for relatively large values of m0 and of m1/2.
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